Question

If a statement is true, prove it. If not, give an example of why it is false. Please neatly and carefully show all necessary work.

4.If f :RR is such that both (0,0) and f(0.v) are continuous at (0,0), then f(x,y) is continuous at (0,0). 5. If f posses all of its directional derivatives at (a, b), then / is differentiable at (a,b). 6. If fr and fy both exist at (a, b), then all other directional derivatives exist at (a,b). 7. If Tein

Answer 1

o wo - cim charo, hshol-10,oo o sino ho in Chorso. b sin orol no ( 1980+ by ginto - cim hocaguisine Bolca8²9 th sinto) - lim erro sine, ho (caso fsinyol = caso sino Duxcaso, sino floo) = (sin ol. sing. = tano. 8ino. pl. Dufle it. Dufloof enist but is not differentiable at (0,0) (6) If tre and fy both exist at carb) then all other directional derivative need not to be enigt. counter example frey) = ( rely if yto Then htoh L 0 ; if yoo je f100) -0. in fire 2010) in f(h.) - 10.0 -0.2 fy(0:0) - test 160,K) - 660,) 0. Die ras ensino £10,0) = lint (huy hu l-f100) Senist oh - lim fchoso, sino to hoh h 80 to cesto Choo - (Piercurso sinopf coo) - = 0 (does not exist), 5 coto b


Answer: G ) then frey) din floc, mrel 1 ich die R is such that both foco) and fro,) are continuous then ſi need not to continuous. cocenter example, a let fray) ay if (rad) + (0,0) co ; if (ah) = 10,0 then fue, o=0 and 100,4)=0. So, it is. continuous at cool Because ein fugl = fcoull I cry+20,0) But sway) is need not to be continuous. Because if we take y=ma. mact 02x2 - itme (ryl Wol. & cemit is depend epon (m), it is not unique Therefore fugl is not continuous at cool (5) It f passes all of its directional derivatives at caub), then t is need not to be differentiable couter example: cet fory) = ( if any) $60,0 lo ifcayl 360,00 if we take a my. Then, lim finch) = myt. (47) Cool 4tmyt 1.4m2 So limit is depends on (m . then it is not unique and we gay frys is not continuous at (0,0). Trovefore, flouh) is not differentiable at 100% But fray) possess all of its divational derivating at o.o. - m